calculations with values with uncertainties, error propagation

error, propagation, uncertainties, uncertainty, calculations, standard, deviation, derivatives, partial, differentiation, autodiff, autodifferentiation, error-propagation
pip install uncertainties==3.2.1



https://readthedocs.org/projects/uncertainties/badge/?version=latest https://pepy.tech/badge/uncertainties/week https://img.shields.io/github/actions/workflow/status/lmfit/uncertainties/python-package.yml?logo=github%20actions

The uncertainties package allows calculations with values that have uncertaintes, such as (2 +/- 0.1)*2 = 4 +/- 0.2. uncertainties takes the pain and complexity out of error propagation and calculations of values with uncertainties. For more information, see https://uncertainties.readthedocs.io/

Basic examples

>>> from uncertainties import ufloat
>>> x = ufloat(2, 0.25)
>>> x

>>> square = x**2
>>> square
>>> square.nominal_value
>>> square.std_dev  # Standard deviation

>>> square - x*x
0.0  # Exactly 0: correlations taken into account

>>> from uncertainties.umath import sin, cos  # and many more.
>>> sin(1+x**2)

>>> print (2*x+1000).derivatives[x]  # Automatic calculation of derivatives

>>> from uncertainties import unumpy  # Array manipulation
>>> varr = unumpy.uarray([1, 2], [0.1, 0.2])
>>> print(varr)
[1.0+/-0.1 2.0+/-0.2]
>>> print(varr.mean())
>>> print(unumpy.cos(varr))
[0.540302305868+/-0.0841470984808 -0.416146836547+/-0.181859485365]

Main features

  • Transparent calculations with uncertainties: Liittle or no modification of existing code is needed to convert calculations of floats to calculations of values with uncertainties.
  • Correlations between expressions are correctly taken into account. Thus, x-x is exactly zero.
  • Most mathematical operations are supported, including most functions from the standard math module (sin,...). Comparison operators (>, ==, etc.) are supported too.
  • Many fast operations on arrays and matrices of numbers with uncertainties are supported.
  • Extensive support for printing numbers with uncertainties (including LaTeX support and pretty-printing).
  • Most uncertainty calculations are performed analytically.
  • This module also gives access to the derivatives of any mathematical expression (they are used by error propagation theory, and are thus automatically calculated by this module).

Installation or upgrade

To install uncertainties, use:

pip install uncertainties

To upgrade from an older version, use:

pip install --upgrade uncertainties

Further details are in the on-line documentation.

Git branches

The GitHub master branch is the latest development version, and is intended to be a stable pre-release version. It will be experimental, but should pass all tests.. Tagged releases will be available on GitHub, and correspond to the releases to PyPI. The GitHub gh-pages branch will contain a stable test version of the documentation that can be viewed at https://lmfit.github.io/uncertainties/. Other Github branches should be treated as unstable and in-progress development branches.


This package and its documentation are released under the Revised BSD License.


This package was created back around 2009 by Eric O. LEBIGOT.

Ownership of the package was taken over by the lmfit GitHub organization in 2024.